In Δ ABC, AD is the bisector of ∠A, meeting side BC at D. If AB = 10 cm, AC = 14 cm, and BC = 6 cm, find the lengths of segments BD and D
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Step-by-Step Solution
Step 1: Understand the Angle Bisector Theorem
The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. In △ABC, AD bisects ∠A, so it divides side BC into segments BD and DC, such that the ratio of AB to AC is equal to the ratio of BD to DC.
Step 2: Substitute given values into the theorem
We are given the lengths of the sides: AB = 10 cm and AC = 14 cm. We substitute these values into the Angle Bisector Theorem equation.
Step 3: Simplify the ratio and express BD in terms of DC
We simplify the ratio 1410 to 75. Then, we can express BD in terms of DC, which will be useful for the next step.
Step 4: Use the total length of BC
We know that the total length of side BC is 6 cm, and it is composed of segments BD and DC. So, BC=BD+DC. We substitute the given value of BC and the expression for BD from the previous step.
Step 5: Solve for DC
Combine the terms involving DC. To solve for DC, multiply both sides by the reciprocal of 712. Simplify the fraction to get the length of DC.
Step 6: Solve for BD
Since BC=BD+DC, we can find BD by subtracting DC from BC. Substitute the known values to find the length of BD.